Classical Topology and Quantization.
Abstract
It is known that there is no unique way to quantize a system if the configuration space Q of the system is topologically nontrivial in the sense that there are loops in Q which can not be smoothly shrunk to a point. We have investigated this quantization ambiguity for certain systems by analysing the topology of their configuration spaces. One class of systems we have considered is that of generally covariant gauge theories in topologically nontrivial space times. It is known in the case of nonabellian gauge theories (for example QCD) in Minkowski spacetime that the topology of the gauge group introduces a free parameter in the theory usually called as the QCD theta angle. When general covariance is included, a certain diffeomorphism group also plays an important role in determining the nature of this parameter. We find that the topology of the diffeomorphism group can get mixed up with the topology of the gauge group in a nontrivial way. This mixing gives rise to a spectrum of generalized theta angles, and for certain manifolds and gauge groups leads to the quantization of these angles as well. The new variable approach to gravity or any approach using vielbeins resembles a gauge theory with diffeomorphism invariance. Thus the nontrivial mixing of the gauge group topology and the topology of the diffeomorphism group can happen for such a theory as well. In particular we find that a spectrum of gravity theta angles or their suitable analogues can appear in a quantum theory of gravity in such approaches. The second system which we have studied is a system of strings in 3 + 1 spacetime dimensions. We have studied those strings which arise for example in phase transitions. We have shown that there are quantum theories associated with a system of such strings with unusual statistical features such as the failure of the normal spinstatistics relation and the possibility of quantization such that the strings are not bosons, fermions or paraparticles.
 Publication:

Ph.D. Thesis
 Pub Date:
 1989
 Bibcode:
 1989PhDT.......212S
 Keywords:

 Physics: Elementary Particles and High Energy; Mathematics